Classification of skew symmetric matrices

Author:
Berndt Brenken

Journal:
Proc. Amer. Math. Soc. **108** (1990), 163-169

MSC:
Primary 15A72; Secondary 15A21

DOI:
https://doi.org/10.1090/S0002-9939-1990-0986646-4

MathSciNet review:
986646

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Abstract | References | Similar Articles | Additional Information

Abstract: The group ${\text {GL(}}d,\mathbb {Z}{\text {) = Aut(}}{\mathbb {Z}^d}{\text {)}}$ acts on the $\mathbb {Z}$-module $\operatorname {Hom} {\text {(}}{\Lambda ^2}{\mathbb {Z}^d},\mathbb {Z}/a\mathbb {Z}){\text {by}}\varphi \to \varphi {\text {(}}\alpha \Lambda \alpha {\text {)}}\quad {\text {(}}\alpha \in {\text {Aut}}{\mathbb {Z}^d}{\text {)}}$. Associated with each $\varphi$ in $\operatorname {Hom} {\text {(}}{\Lambda ^2}{\mathbb {Z}^d},\mathbb {Z}/a\mathbb {Z})$ is a finite set of invariants completely describing the orbit of $\varphi$ under this action. The result holds with $\mathbb {Z}$ replaced by an arbitrary commutative principal ideal domain.

- Berndt Brenken,
*A classification of some noncommutative tori*, Proceedings of the Seventh Great Plains Operator Theory Seminar (Lawrence, KS, 1987), 1990, pp. 389–397. MR**1065837**, DOI https://doi.org/10.1216/rmjm/1181073114
F. G. Frobenius, - Morris Newman,
*Integral matrices*, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 45. MR**0340283**

*Theorie der linearen Formen mit ganzen coefficienten*, J. Reine Angew. Math.

**86**(1880), 96-116.

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Keywords:
Module,
principal ideal domain,
pfaffian,
skew symmetric matrix,
automorphism

Article copyright:
© Copyright 1990
American Mathematical Society